4 Subscripts and braces

This is a comment to Sunil's answer on Euclid's proof of the infinitude of primes but I don't have enough point to leave a comment.

There is a generalization of Euclid's proof which should be well known but I seldom see it mentioned. If $a_1,...,a_r$ are pairwise relatively prime integers, then for any subset $I \subset {1,...,r}$, \{1,...,r\}$,$a(r+1)=\prod_{i a_{r+1}=\prod_{i \in I}a_i +\prod_{i \not\in I}a_i$, is relatively prime to all the$a1,...,an$.a_1,...,a_r$.

3 Fixed notation

This is a comment to Sunil's answer on Euclid's proof of the infinitude of primes but I don't have enough point to leave a comment.

There is a generalization of Euclid's proof which should be well known but I seldom see it mentioned. If $a_1,...,a_r$ are pairwise relatively prime integers, then for any subset $I \subset {1,...,r}$, $a(r+1)=\prod_{i \in I}a_i +\prod_{i \not in I}ai$not\in I}a_i$, is relatively prime to all the$a1,...,an$. 2 deleted 559 characters in body This is a comment to Sundil's Sunil's answer on Euclid's proof of the infinitude of primes but I don't have enough points point to leave a comment. There is a generalization of Euclid's proof which should be well known but I seldom see it mentioned. If$p1,...,pr$are distinct primes, then for any subset$I \subset {1,...,r}$, and$P=\prod_{i \in I}pi + \prod_{i \not in I}pi$, (where empty product is 1) then$P$is relatively prime to all the pi and hence must contain a prime divisor different from all the$pi$. Sunil's addition of a multiple is new to me and one can include this as$P=m\prod_{i \in I}pi + n\prod_{i \not in I}pi$, where m is relatively prime to pi for i \not in I and n is relatively prime to pi for i \in I. The same argument also implies by looking at prime divisor that if$a_1,...,a_r$are pairwise relatively prime integers, then for any subset$I \subset {1,...,r}$,$a(r+1)=\prod_{i \in I}a_i +\prod_{i \not in I}ai$, is relatively prime to all the$a1,...,an\$.